A minimum reversing set of a digraph is a smallest sized set of arcs which when reversed makes the digraph acyclic. We investigate a related issue: Given an acyclic digraph D, what is the size of a smallest tournataent T which has the arc set of D as a minimum reversing set? We show that such a T always exists and define the reversing number ofan acyclic digraph to be the number of vertices in ...

An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. It is proved that for every integer sX2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree of each vertex v is at most degðvÞ s 1 l m : For all digraphs this degree bound is optimal. r...

Let D be a digraph and X ~ V(D). By pushing X we mean reversing the orientation of each arc of D with exactly one end in X. Klostermeyer proved that it is NP-complete to decide if a given digraph can be made acyclic using the push operation. By contrast, Huang, MacGillivray, and Wood showed that the problem of deciding if a given multipartite tournament can be made acyclic using the push operat...

A digraph D is connected if the underlying undirected graph of D is connected. A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order n. Connected convex subgraphs of connected acyclic digraphs are of int...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely n-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in...

Given a digraph D = (V ,A) and an X ⊆ V , DX denotes the digraph obtained from D by reversing those arcs with exactly one end in X. A digraph D is called acyclically pushable when there exists an X ⊆ V such that DX is acyclic. Huang, MacGillivray and Yeo have recently characterized, in terms of two excluded induced subgraphs on 7 and 8 nodes, those bipartite permutation digraphs which are acycl...

Given an embedded planar acyclic digraph G, we define the problem of acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM) to be the problem of determining a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to an acyclic hamiltonian digraph. Our results include: ...

A minimum feedback arc set of a digraph is a smallest sized set of arcs that when reversed makes the resulting digraph acyclic. Given an acyclic digraph D, we seek a smallest sized tournament T that has D as a minimum feedback arc set. The reversing number of a digraph was defined by Barthélemy et. al. to be r(D) = |V (T )| − |V (D)|. We will completely determine the reversing number for a disj...

In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph G, determine whether there exists an upward 2-page book embedding of G preserving the given planar embedding. Given an embedded ...